Thanks for the response - I didn't mean to ignore it, I was just busy with flooring the other part of the house! I took your advice and ripped up some carpet to see what I was working with last night.
It looks fairly promising -- the risers are in good shape, and they don't stop short of the skirtboard; in fact, they pass behind it, although there is still a bit of a gap as you can see from the second picture. How does this plan sound?
1) Cut overhang off treads
2) Sand the risers
3) Use drywall mud to go over the nail indentions on the risers
4) Use drywall mud to fill in the gaps between the riser and the skirtboard (see pics)
5) Prime & paint risers with white glossy (also repaint the skirtboards!)
6) Install flooring + nosing on treads
As for using 1-piece treads, well, I have already ordered the 3.25" strip flooring and nosings, so that's what I'll have to use. In hindsight, I would have tried to find 1-piece treads.